June  2015, 5(2): 359-376. doi: 10.3934/mcrf.2015.5.359

Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

1. 

GE Global Research, 1 Research Circle, Niskayuna, NY 12309, United States

2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074

3. 

Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Received  January 2014 Revised  February 2014 Published  April 2015

To treat networked systems involving uncertainty due to randomness with both continuous dynamics and discrete events, this paper focuses on diffusions modulated by a continuous-time Markov chain. In our paper [19], we considered ordinary differential equations with Markovian switching. This paper further treats more complex cases, namely, stochastic differential equations with Markovian switching. Our goal is to stabilize the systems under consideration. One of the difficulties is that the systems grow much faster than the allowable rates in the literature of stochastic differential equations. As a result, the underlying systems have finite explosion time. To overcome the difficulties, we develop feedback controls to extend the local solutions to global solutions and to stabilize the resulting systems. The feedback controls are Brownian type of perturbations. We establish the existence of global solution, prove the stability of the resulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficient conditions for almost sure stability. Then we present numerical examples to illustrate the main results.
Citation: Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359
References:
[1]

J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations,, Systems & Control Letters, 54 (2005), 1069.  doi: 10.1016/j.sysconle.2005.03.003.  Google Scholar

[2]

J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise,, IEEE Trans. Automat. Control, 53 (2008), 683.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[3]

L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise,, SIAM J. Control Optim., 21 (1983), 451.  doi: 10.1137/0321027.  Google Scholar

[4]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364.  doi: 10.1016/j.jmaa.2003.12.004.  Google Scholar

[5]

F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth,, Systems & Control Letters, 57 (2008), 262.  doi: 10.1016/j.sysconle.2007.09.002.  Google Scholar

[6]

M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1952.  doi: 10.1137/S0363012996299302.  Google Scholar

[7]

R. Z. Khasminskii, Stochastic Stability of Differential Equations,, $2^{nd}$ edition, (2012).  doi: 10.1007/978-3-642-23280-0.  Google Scholar

[8]

R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions,, J. Differential Eqs., 161 (2000), 154.  doi: 10.1006/jdeq.1999.3647.  Google Scholar

[9]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, $2^{nd}$ edition, (2003).  doi: 10.1007/b97441.  Google Scholar

[10]

B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,, J. Math. Anal. Appl., 339 (2008), 419.  doi: 10.1016/j.jmaa.2007.06.058.  Google Scholar

[11]

R. Liptser and A. N. Shiryaev, Theory of Martingale,, Kluwer Academic Publishers, (1989).  doi: 10.1007/978-94-009-2438-3.  Google Scholar

[12]

X. Mao, Stability of stochastic differential equations with Markovian switching,, Stochastic Process. Appl., 79 (1999), 45.  doi: 10.1016/S0304-4149(98)00070-2.  Google Scholar

[13]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/9781860948848_fmatter.  Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008).  doi: 10.1533/9780857099402.  Google Scholar

[15]

A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations,, Amer. Math. Soc., (1989).   Google Scholar

[16]

F. Wu and S. Hu, Suppression and stabilisation of noise,, Internat. J. Control, 82 (2009), 2150.  doi: 10.1080/00207170902968108.  Google Scholar

[17]

G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions,, SIAM J. Math. Anal., 41 (2010), 2335.  doi: 10.1137/080727191.  Google Scholar

[18]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).  doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[19]

G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM J. Appl. Math., 72 (2012), 1361.  doi: 10.1137/110851171.  Google Scholar

[20]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154.  doi: 10.1016/j.jmaa.2009.03.066.  Google Scholar

show all references

References:
[1]

J. A. D. Appleby and X. Mao, Stochastic stabilisation of functional differential equations,, Systems & Control Letters, 54 (2005), 1069.  doi: 10.1016/j.sysconle.2005.03.003.  Google Scholar

[2]

J. A. D. Appleby, X. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise,, IEEE Trans. Automat. Control, 53 (2008), 683.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[3]

L. Arnold, H. Crauel and V. Wihstusz, Stabilization of linear system by noise,, SIAM J. Control Optim., 21 (1983), 451.  doi: 10.1137/0321027.  Google Scholar

[4]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364.  doi: 10.1016/j.jmaa.2003.12.004.  Google Scholar

[5]

F. Deng, Q. Luo, X. Mao and S. Pang, Noise suppresses or expresses exponential growth,, Systems & Control Letters, 57 (2008), 262.  doi: 10.1016/j.sysconle.2007.09.002.  Google Scholar

[6]

M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1952.  doi: 10.1137/S0363012996299302.  Google Scholar

[7]

R. Z. Khasminskii, Stochastic Stability of Differential Equations,, $2^{nd}$ edition, (2012).  doi: 10.1007/978-3-642-23280-0.  Google Scholar

[8]

R. Z. Khasminskii and G. Yin, Asymptotic behavior of parabolic equations arising from null-recurrent diffusions,, J. Differential Eqs., 161 (2000), 154.  doi: 10.1006/jdeq.1999.3647.  Google Scholar

[9]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications,, $2^{nd}$ edition, (2003).  doi: 10.1007/b97441.  Google Scholar

[10]

B. Lian and S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,, J. Math. Anal. Appl., 339 (2008), 419.  doi: 10.1016/j.jmaa.2007.06.058.  Google Scholar

[11]

R. Liptser and A. N. Shiryaev, Theory of Martingale,, Kluwer Academic Publishers, (1989).  doi: 10.1007/978-94-009-2438-3.  Google Scholar

[12]

X. Mao, Stability of stochastic differential equations with Markovian switching,, Stochastic Process. Appl., 79 (1999), 45.  doi: 10.1016/S0304-4149(98)00070-2.  Google Scholar

[13]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/9781860948848_fmatter.  Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008).  doi: 10.1533/9780857099402.  Google Scholar

[15]

A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations,, Amer. Math. Soc., (1989).   Google Scholar

[16]

F. Wu and S. Hu, Suppression and stabilisation of noise,, Internat. J. Control, 82 (2009), 2150.  doi: 10.1080/00207170902968108.  Google Scholar

[17]

G. Yin, X. R. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: Numerical algorithms and existence and uniqueness of solutions,, SIAM J. Math. Anal., 41 (2010), 2335.  doi: 10.1137/080727191.  Google Scholar

[18]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).  doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[19]

G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM J. Appl. Math., 72 (2012), 1361.  doi: 10.1137/110851171.  Google Scholar

[20]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154.  doi: 10.1016/j.jmaa.2009.03.066.  Google Scholar

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